The notion of there being a Euclidean Space begins to make sense after you encounter reasons to develop and understand the alternatives, such as Minkowski Space. The intent here is to introduce euclidean space in a manner so that we can frequently compare it to Minkowski space.
A Euclidean space is a finite dimensional vector space over the real numbers and it has an inner product.
There is a Euclidean Space for each dimension.
- We have a 2-tuple for two dimensional Euclidean space (x,y)
- We have a 3-tuple for three dimensional Euclidean space (x,y,z)
In a Minkowski space, there is a Minkowski inner product that calculates the spacetime interval between two events.
What is an event? Let’s give an example: a blue flash occured on your Christmas tree and we report it as four numbers , three giving location in space and one giving location on a time line.
Appendix A
A space and a coordinate system are two different things.
There is more than one coordinate system that can be used in Euclidean Space. Both Cartesian coordinates and polar coordinates are valid.
Cartesian Product
The Cartesian Product of real numbers gives the n-tuple needed for a Euclidean Space. You were working with Euclidean spaces when you worked with either (x,y) or (x,y,z).
For a 2-tuple:
example: (x,y)
For a 3-tuple:
example: (x,y,z)
(x,y) could be the point at (x,y) or it could be a vector from (0,0) to (x,y).
Thinking about vectors is nice because we can add vectors together: (1,1) + (2,3) = (3,4). We can’t draw a picture for adding points. However, it should make sense that you can add two equal length lists of numbers together (and the result will be another list).
We will look at an example where there are two dimensions, 
If 
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